%\include {preamble}

%\begin{document}


\frame{ \frametitle{Semi-supervised Domain Adaption}
  Consider some with some labels in the target domain.
  Let $\tilde{X} = \tilde{X}_l \bigcup \tilde{X}_u$, where $\tilde{X}_l=\{\tilde{x}_{li}\}^{N_{l2}}_{i = 1}$ has labels,

 say $\{ \tilde{y}_{li} \}^{N_{l2}}_{i = 1}$,

 $\tilde{X}_l=\{\tilde{x}_{ui}\}^{N_{u2}}_{i = 1}$ ($N_{l2} + N_{u2} = N_2$).

 We now use a $dN' \times (N_{l1} + N_{l2})$ data matrix and the corresponding $(N_{l1} + N_{l2}) \times 1$ label vector to build classifer D.

}



\frame{ \frametitle{Adaption Across Multiple Domains}
  For the scenarios with  multiple domains, say, $k_1$ source domains and $k_2$ target domains,

  \begin{itemize}
    \item Create generative subspaces $S_{11}, S_{12}, \ldots, S_{1k_1}$ corresponding to the source,
    and $S_{21}, S_{22}, \ldots, S_{2k_2}$ for the target

    \item Compute the mean of the subspaces, say $\bar{S}_1$, and the mean for target $\bar{S}_2$

    \item Create the intermediate subspaces between $\bar{S}_1$ and $\bar{S}_2$

    \item Learn the classifier $D$ as before.

  \end{itemize}

}

\frame{ \frametitle{Karcher Mean}
  \begin {enumerate}
    \item Given a set of $k$ points $\{ q_i \}$ on the manifold
    \item Let $\mu_0$ be an initial estimate of the     \hyperlink{KarcherMean}{\beamergotobutton{Karcher mean}}
 by picking one of $\{q_i\}$ at random
    \item For each $i = 1, \ldots, k$, compute the inverse exponential map $\nu_i = \exp^{-1}_{\mu_j}(q_i)$
    \item Compute the average tangent vector $\bar{\nu} = \frac{1}{k} \sum^{k}_{i=1}\nu_i$
    \item If $||\bar{\nu}||$ is small, then stop. Else, move $\mu_j$ in the average tangent direction using $\mu_{j+1} = \exp_{\mu_j}(\epsilon \bar{\nu})$, where $\epsilon > 0$ is small step size, typically 0.5
    \item Set j = j+1 return to Step 3.
  \end {enumerate}
}


\frame{ \frametitle{Karcher Mean}
  \begin {figure}
  \tiny
    \begin {tabular} {cc}
    \includegraphics [width = 0.4\textwidth] {fig/demo_km/1_points} &
    \includegraphics [width = 0.4\textwidth] {fig/demo_km/2_init_mean} \\
    (1)Points on Grassmann manifold & (2) Select the initial Karcher Mean $\mu_0$ \\
    \includegraphics [width = 0.5\textwidth] {fig/demo_km/3_tan_plane} &
    \includegraphics [width = 0.5\textwidth] {fig/demo_km/4_expmap} \\
    (3)Tangent plane of $\mu_0$ & (4) Compute the inverse exp. map for each point
    \end {tabular}
  \end {figure}
}


\frame{ \frametitle{Karcher Mean}
  \begin {figure}
    \tiny
    \begin {tabular} {cc}
    \includegraphics [width = 0.5\textwidth] {fig/demo_km/5_tanvec} &
    \includegraphics [width = 0.5\textwidth] {fig/demo_km/6_avgtan_vec} \\
    (5)Compute the tangent vectors & (6) Average the tangent vectors $\bar{\nu}$ \\
    \includegraphics [width = 0.5\textwidth] {fig/demo_km/7_invexpmap} &
    \includegraphics [width = 0.5\textwidth] {fig/demo_km/8_mov} \\
    (7) Compute the exp. map of $\bar{\nu}$ & (8) Update the $\mu_0$
    \end {tabular}
  \end {figure}
}


%\end{document}

